• Open Access

Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes

Theodore J. Yoder, Ryuji Takagi, and Isaac L. Chuang
Phys. Rev. X 6, 031039 – Published 13 September 2016

Abstract

It is an oft-cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal. This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation. Widely overlooked, however, is the possibility of nontransversal, yet still fault-tolerant, gates that work directly on small quantum codes. Here, we demonstrate precisely the existence of such gates. In particular, we show how the limits of nontransversality can be overcome by performing rounds of intermediate error correction to create logical gates on stabilizer codes that use no ancillas other than those required for syndrome measurement. Moreover, the logical gates we construct, the most prominent examples being Toffoli and controlled-controlled-Z, often complete universal gate sets on their codes. We detail such universal constructions for the smallest quantum codes, the 5-qubit and 7-qubit codes, and then proceed to generalize the approach. One remarkable result of this generalization is that any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of fault-tolerant gates. Another is the interaction of logical qubits across different stabilizer codes, which, for instance, implies a broadly applicable method of code switching.

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  • Received 12 March 2016

DOI:https://doi.org/10.1103/PhysRevX.6.031039

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & Technology

Authors & Affiliations

Theodore J. Yoder, Ryuji Takagi, and Isaac L. Chuang

  • Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Popular Summary

Building something perfect from imperfect components is common to all fields of science and engineering. However, in quantum computing, this idea truly is fundamental. A standard procedure for protecting fragile quantum bits involves using entanglement in a technique called quantum coding, wherein the extra degrees of freedom in an extended system are periodically measured and used to diagnose errors on a protected subsystem. It is also necessary to compute directly on quantum codes while tolerating faulty gates. Unfortunately, there are theorems that dictate that the easiest route to fault-tolerant quantum computation, called transversality, will fail. Researchers accordingly appear to be all but resigned to using exceedingly large codes and/or ancilla qubits numbering in the tens to hundreds. Here, using a general and generalizable method, we show that universal fault-tolerant computation is possible using even the smallest quantum code and no additional qubits other than those necessary to correct errors.

We focus on 5- and 7-qubit codes and correct errors within the circuit to inhibit the uncontrolled propagation of errors. The circuits that we study can recover from a single fault without triggering a logical error. We show that fault-tolerant gates that are also nontransversal are possible, meaning a qubit from one code block may interact with multiple qubits from another code block. Moreover, we show that our approach succeeds in bestowing universality on many other quantum codes, greatly expanding the set of useful quantum codes. However, we do note that our investigation is limited to concatenated codes.

We expect that our findings will pave the way for investigations of more complex circuits that can recover from two or more faults without causing a logical error.

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Vol. 6, Iss. 3 — July - September 2016

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